Mohamed Elyounoussi and 17-year-old striker Viktor Dadason scored on either side of the break, before Jordan Larsson converted a penalty towards the end of the match.
Thanks to the win, FC Copenhagen has a solid grip on seventh place, which gives them access to a European playoff spot. They have a four-point gap to OB with three rounds remaining in the Danish Superliga.
Vejle, however, began the match in promising style, and they thought they had taken the lead after 19 minutes when Wahid Faghir steered a header into the back of the net from a narrow angle, but the goal was disallowed for offside.
A few minutes later, Elyounoussi wasted a great chance before Andreas Cornelius was denied by the post. However, after 31 minutes, the home side brought the local spectators to their feet when Elias Achouri searched into the box and produced an accurate delivery for Elyounoussi, who fired the ball through the legs of Vejle-keeper Tobias Jakobsen.
Despite being already relegated, Vejle stood their ground well throughout the match, but after 73 minutes, the match was in reality decided when Jordan Larsson escaped a defensive challenge to set up the onrushing Viktor Dadason, who doubled the lead.
FC Copenhagen then put the icing on the cake six minutes from the end when Larsson converted a penalty after Max Jensen was adjudged to have touched the ball with his arm in the box.